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Network Entropy and the Cancer Cell
Local and global cancer hallmarks
and applications
Contents
1.
2.
3.
4.
Cancer research in the 21st century
Entropy in gene expression
Entropy in copy number
Conclusions
1. Cancer research in the 21st century
Three reasons why we haven’t cured cancer
1. Cancer consists of an unknown number of subtypes whose origins and conditions are not well
known: which genes and cellular functions are
important?
Daily Mail
BBC
Three reasons why we haven’t cured cancer
2. A given tumor tends to consist of many different
sub-populations. Thus any treatment tends to be
like cutting off some of the heads of a hydra.
A relatively minor subpopulation of tumor cells survives chemotherapy and
arises to become the dominant sub-type at relapse
Ding et al, Nature 2012
Three reasons why we haven’t cured cancer
3. Most often cancer is driven by alterations in a
complex gene network. The biological picture of
the network state is still largely incomplete.
In ordinary
tumors with
RAF activated
you only need
to inhibit MEK
to put the
cancer into
remission.
But when RAS
is mutant, you
have to inhibit
PI3K and MEK!
In this case knowing the transduction pathway involved gives
biological understanding to why treatments work.
Unfortunately, these are not known for most pathways! Engelmann et al, Nature Med. 2008
Modern tools in cancer genomics
• We now have the technology to study cancer at
the level of entire genomes:
Copy
Gene
number
expression
Finding cancer genes: the traditional approach
• Given genomic disease data it is natural to ask
which parts of the genome are different in the
disease and how that has affected the biological
Rank genes according to
function of the system.
Check to see
which known
biological
functions of the
cell contain
significantly
many of these
genes. Control
for false
discovery!
size of change and select
those such that only 5%
are expected to be false
positives.
The network
structure is ignored
in this approach
Network approaches in cancer
• It is now known that most cancers are not caused by
malfunctions in a single protein.
• In fact most tumors are characterized by hundreds of
alterations (copy number, DNA methylation, …), often of
different unknown levels of importance (drivers vs
passengers).
• To address this challenge, biological data are often
combined with interaction network models, giving rise to
so-called integrated approaches.
The protein interaction network (PIN)
• The approximately 20,000 genes in the human genome are
synthesized into proteins. These proteins interact, although we believe
ourselves only to know a small proportion of all interactions; perhaps
only 10%.
• The current interaction network models are obtained from
agglomerating results of vast amounts of experiments.
• http://www.pathwaycommons.org/
It is believed that around 50%
of the interactions in the yeast
protein interaction network are
known.
Integrated expression-PIN networks
• Proteins that interact are in general more
correlated at the expression (mRNA) level:
Example: normal bladder tissue
(around 40 samples)
Integrated expression-PIN networks in cancer
Normal gene
expression
PINs (knowledge & predicted)
Experiment
Domain Analysis
Homology
(common
ancestry)
Cancer gene
expression
Which local and global
network properties best
define the cancer
phenotype?
Normal network
Cancer network
Advantages of network approaches
• Improved interpretation: networks provide a biological context to
interpret data like gene expression and copy number. For example, the
network approach has uncovered subnetworks whose individual
properties help to make predictions about cancer.
Taylor IW Nat Biotech 2009
• Noise reduction: using the
eigendecomposition of the graph
Laplacian may help remove noise in
the data.
Rapaport BMC Bioinformatics 2007
• Improved prognostic models:
network allows more robust
identification of relevant prognostic
markers.
Chuang Mol Syst Bio 2007
Incoherency of unsupervised prediction
Teschendorff BMC Syst Bio 2010
Data most
coherent with
75% of
eigenvalues
removed
Similar
random
networks
% eigenvalues removed
A network approach to improving classification
• Chuang et al used the mean expression levels of
subnetworks of the PIN as features in a classifier to predict
whether breast cancers would spread (metastasize).
Subnetwork markers
Area-Under-Curve performance
Classifier
Build classifier
(mean
expression levels
of subnetwork
markers)
Test 1
Test 2
Chuang et al, Mol Syst Bio, 2007
Properties of individual genes
• Unsurprisingly, simple topological properties of proteins in the PIN
have been associated with function.
• In yeast, it was found that betweenness is a more effective indicator of
“essentiality” (i.e. the organism either dies or ceases growth without it)
than degree.
The protein CAK1P
in yeast, an
essential, but lowdegree protein of
high betweenness.
CAK1P
Gerstein et al, PLoS Comp Bio 2007
Properties of individual genes
• By representing the structure of the network near a protein
in terms of some of the different possible sub-networks
containing it (viewed as sub-networks of the PIN), it is
possible to predict the protein’s function.
=
Protein of
interest
Pick a subset of the
set of networks
(graphs)
Compute the number of
copies of each of these
networks that include your
protein of interest
5
2
8
2
0
5
Classifier
Predict what sort
of thing the
protein does: e.g.
aid in killing the
cell, fix the DNA,
communicate
between cells,
etc.
Kuchalev et al, J R Soc Interface 2010
Dynamics on the PIN
• Given weights on either the vertices (genes) or edges
(interactions) in the PIN, there are several natural
constructions giving rise to random walks.
• Komurov used the invariant measures (after adjustment) of
walks coming directly from the expression-PIN to
understand response to DNA damage.
Invariant measure
re-scaled to avoid
great bias
towards hubs
Komurov et al, PLoS Comp Bio 2010
2. Entropy in Gene Expression
Motivation
• Want to find a new framework to find genes responsible for “driving”
cancers.
• At present rather little is known that distinguishes generic normal
tissue cells and cancerous cells at the systems level, in particular of
protein-interactions and expression (mRNA) levels.
• We investigate this by looking to see if changes in the information
content across various notions of what might be called “molecular
entropy” to distinguish normal and cancer tissues.
Normal lung tissue
Cancerous lung tissue
A random walk on the PIN
• Given interacting proteins i and j, their correlation
Cij across samples of a given phenotype gives a
weight on the edge between them.
• Scaling this to be non-negative, as wij = (1+ Cij)/2
(or perhaps |Cij|), we obtain a random walk on the
PIN by normalizing to obtain a probability pij of
walking from i to j
The choice of walk
• With the choice of edge weight wij = (1+ Cij)/2 we force the
random walk to flow like information would flow through
the system; in the direction of signal transduction paths
and away from inactive or possibly inhibitory interactions.
• The choice of edge weight wij = |Cij| is also valid, but treats
inhibition and activation equally.
Information flow in
direction of signal
Flow blocked by
inhibitory interaction
Molecular entropy
• Observe that the walk probabilities emanating
from a vertex (protein) gives a probability
distribution corresponding to that gene.
• We refer to the Shannon entropy of this, i.e.
as the molecular flux entropy of the protein i.
• The Shannon entropy in a sense measures the
predictability of the walk: on a finite set of values,
a uniform distribution is least predictable, and a
distribution always taking the same value the
most.
Information loss in the cancer cell
• Significant increases were observed in the transitions from
normal to cancerous to metastatic cancers (those which
have spread). This was observed to be the case across
several cancer types.
Non-metastatic
Metastatic (cancer has spread)
Gene-1
inactivated in
metastasis
leads to loss
of correlations
and greater
entropy
Teschendorff & Severini BMC Systems Biology 2010
Information loss in the cancer cell
• Compared to the average local correlation, (i.e.
the mean correlation of a node with its nearest
neighbors) and the mean local absolute
correlation, the molecular entropy was a better
distinguisher of normal, cancer and metastatic
cancers.
Information loss in the cancer cell
• As a sanity check: the Shannon entropy was also
higher in randomised null networks in which the
expression values were randomly permuted
among nodes in the PIN.
Entropy is higher in
primary breast cancer
tumors that have
spread (metastasized)
Teschendorff & Severini BMC Systems Biology 2010
Dynamical entropy over longer distances
• The previous notion of molecular entropy obtained
weights from the correlations in two different
phenotypes and computed the disorder of these
(after slight adjustment) looking only at immediate
neighbors.
• Observe that the kth power of the stochastic
matrix p = (pij) gives the information flow over
distances of length k.
• The total information flow of various distances
between two genes can be obtained by taking a
choice of linear combination of powers of p.
A natural combination of powers
• Motivated by statistical physics, we introduce a
temperature parameter t and consider the family
of matrices K(t) obtained as
• This series converges and satisfies a modified
heat diffusion equation
Heating it up
• In the “hot” temperature limit, this indeed
approximates a solution of the heat diffusion
equation
• For each t we may write down a “global” entropy
S(t) of the information flux as
where Q is the number of non-zero entries of K.
• We refer to this as the global flux entropy.
West et al Submitted 2012
Covariance entropy
• Before studying the flux entropy further, we
introduce another notion recently studied.
• The covariance entropy quantifies the degree of
similarity between samples as determined by their
Pearson correlations, first considered in the
context of gene expression by van Wieringen and
van der Vaart.
• They argue that with the accumulation of copy
number aberrations in cancer, there should be an
increase in genomic entropy. For this to be
cancerous, this should be reflected in expression.
van Wieringen et al Bioinforrmatics 2011
When does entropy in copy number lead to
entropy in expression?
DNA copy number
X
S(N)
<
S(C)
?
mRNA expression Y = g(X)
S(N)
Entropy in normal
phenotype
<
S(C)
Entropy in cancer
phenotype
Proposition:
Van Wieringen et al Bioinformatics 2010
Computing covariance entropy
• Modelling gene expression profiles across n
samples of g genes as a multivariate normal Y ~
N(μ,Σ), the entropy H(Y) is given by
• Letting Σi be the covariance matrix restricted to
gene i leads to the local covariance entropy:
A caveat
• The monotone relationship between copy number
X and gene expression Y = g(X) may be severely
compromised if the measured expression levels
are over a mixture of tumour cells and stromal
(non-tumor cells).
Normal
Cancer
Normal
copynumber
Epithelial cell
Immune cell
Epithelial cell,
gene amplified
Low expression
High expression
Normal
Cancer
Global information entropy increases
Flux entropy is increased
in cancer relative to
normal tissue.
Covariance entropy
changes inconsistent.
Maximum difference
attained for paths in the
PIN up to length 3.
West et al Submitted 2012
Correlation path length scales
Local correlations
stronger in every
case.
Natural correlation
length scale on the
mRNA-PIN is 3.
West et al Submitted 2012
Local flux and covariance entropies
Local flux
entropies
increased in
cancer.
Local
covariance
entropies not
always
increased.
Consistent and significant increases in local flux entropy found from
normal to cancer tissue.
West et al Submitted 2012
Differential expression and entropy
• Hypothesis: alterations in genes “driving” the
cancer should lead to disruptions in local and
global gene expression patterns causing changes
in entropy.
• Consequences:
• Genes whose inactivation confers selective
advantage (tumor suppressors) should tend to
show increases in local flux entropy.
• Genes that become activated in cancer
(oncogenes) can be expected to show
reductions.
Verifying changes in expression
• Genes significantly over-expressed in cancer (P < 0.05)
show reductions in flux entropy when compared to genes
which were under-expressed.
• Binding of IGFBP7 to
IGFBP7
•
•
IGF1
Non-metastatic
Metastatic (cancer has spread)
Increase in entropy
IGF1 is reduced and
there is reduced
suppression of VEGFA.
This leads to increased
angiogenesis (growth
of new blood vessels; a
hallmark of cancer) in
the metastatic
phenotype.
IGFBP7 is putative
tumor suppressor;
lower expression in
cancer. Wajapeyee et al Cell
2008, Oh Y et al J Biol Chem 1996
Teschendorff & Severini BMC Systems Biology 2010
Interpreting flux entropy
• The increased flux entropy in cancer may endow
cancer cells with the flexibility to adapt to the strong
selective pressures of the tumor microenvironment.
• The fluctuation theorem of Demetrius et al asserts
ΔRΔS > 0
i.e. there is a correlation between changes in network
entropy S and robustness R. As such it is possible that
cancer alterations leading to significant increases in
flux entropy may contribute to the dynamical
robustness of such cancer cells.
Demetrius et al 2004 Physics A 346
Therapeutic applications
• A great problem is that important cancer-related genes are not directly
druggable.
• In these cases it may be possible to use differential flux entropy to
identify neighboring viable drug targets that also exhibit significant
reductions in flux entropy.
• This computational strategy could therefore guide certain therapeutic
strategies that aim to select drug targets within the same oncogenic
pathway.
?
Often genes are not
druggable because there
are too many similar
genes and drugs tend to
affect all of them.
3. Entropy in copy number
Motivation
• In this section we use a “combinatorial smoothing”
obtained by associating a network known as a
visibility graph to copy number data viewed as a
time series
Visibility
Approx.
entropy
algorithm
Copy
number
Visibility
graph
Information
about
phenotype
Approximate entropy of networks
• Much practical work has considered an entropic
origin for various properties of complex networks,
including biodiversity in ecological networks, the
emergence of degree-degree correlations and
communities in social and biological networks.
• These approaches focus mainly on the ubiquitous
Shannon entropy, and to help complete the
picture we introduce a notion based around the
approximate entropy.
Approximate entropy
• The approximate entropy considered by Pincus
exists as a finite-sized statistic of the EckmannRuelle entropy proposed to measure the
complexity of a system with time evolution.
• The approximate entropy ApEn(m,r,N) of a time
series u of length N is constructed from the image
Xm of the Takens map x(t) = u((t),…,u(t+m–1)).
With
define
The Rukhin estimate
• Intuition is gained due to the combinatorial
interpretation of approximate entropy, due to
Rukhin: given a sequence u of length N on S
symbols {0, 1, …, S – 1}, let ν(I) be the frequency
with which block I occurs. Denoting
the estimate is
• Fact: this is a.s. a good approximation of the
ordinary approximate entropy of u.
Rukhin, J Appl Probab 2000
Upshot of the Rukhin estimate
• As such, on a finite set of symbols, the
approximate entropy adds something new,
measuring something distinct from the ordinary
statistical moments (mean, variance, … ) and the
Shannon entropy of the sequence.
• It can be (almost surely!) thought of measuring
how much data a choice of universal data
compressor would use to store the object.
Approximate entropy of a degree sequence
• As such for a network, we introduce the slide entropy
which is the approximate entropy of a binary sequence
encoding something like an “infinitesimal” disorder of the
finite degree sequence.
• The slide construction is simple: draw the degree
distribution as a partition diagram and trace along it from
left to right. When you go horizontally one unit write down
“0” and vertically one unit, write down “1”.
0
1
0
0
0
1
01000101
0
1
Analytics for the slide entropy
• In nice circumstances (almost surely!)
approximate entropy of this slide sequence is
recovered asymptotically by a Shannon entropy
relating the distribution of 0s and 1s in the
associated sequence, yielding an analytic formula
of empirically measured reasonable accuracy, say
for Poisson networks.
• Almost surely isn’t always…
West et al, Phys Rev E 2012
Visibility graphs
• Visibility graphs are networks associated to time
series that capture features of the time series in
their topology.
Time series
Visibility
algorithm
Visibility
Constructing a horizontal visibility graph
graph
Lacasa & Toral, Phys Rev E 2010
An example application of visibility graphs
• A fractional Brownian motion process of Hurst
exponent H gives rise to a scale free visibility
graph of scale free parameter 3 – 2H.
Fractional Brownian motion (H).
Degree distribution of
associated visibility graph
Lacasa et al, arXiv:0901.0888v1 2009
Approximate entropy of visibility graphs
Uncorrelated white noise gives
rise to visibility graphs of
maximal ApEn (it was known
previously also to maximize
the Shannon entropy of the
degree distribution).
The ApEn of visibility graphs
associated to chaotic maps
reaches a non-zero value,
reminiscent of the underlying
attractor of the dynamics.
As such, the network structure
also inherits the complexity of
the time series.
West et al, Phys Rev E 2012
Approximate entropy of copy number data
Significant
increases in
approximate
entropy of
slide
sequence .
Is the cancer over-expressing
estrogen receptors?
How fast growing is the
cancer?
Has the cancer spread?
N.B. ER-status, Cancer GRADE and Distal Metastasis all correlate in
cancers, but remain distinct phenotypic properties.
West et al, Phys Rev E 2012
4. The End
Summary
1. The flux entropy combines network information with
gene expression to provide a hallmark of cancer.
2. It is a more consistent indicator than more basic
considerations of the correlations, and similarity of
samples even when restricted to the network.
3. It may be useful in guiding therapeutic target selection
and helps to indicate genes driving cancers.
4. The approximate entropy of the slide sequence
associated to copy number data might help to distinguish
tumor grade and metastasis.
References
1. “On dynamical network entropy in cancer”.
James West, Ginestra Bianconi, Simone
Severini and Andrew E. Teschendorff.
arXiv:1202.3015v1 [q-bio.MN].
2. “Approximate entropy of network
parameters”, Physical Review E 2012.
James West, Lucas Lacasa, Simone Severini
and Andrew E. Teschendorff.
arXiv:1201.0045v1 [cond-mat.dis-nn].
Acknowledgements
Andrew Teschendorff
Simone Severini
Lucas Lacasa
Ginestra Bianconi